Let the language $\mathcal L_1^{=}$, with the following set of non-logical symbols {a, b, H, K}, where a, b are individual constants and H, K are 1-ary predicates, and a classic semantics, prove the following statements.
a) The (closed) formula((Ha $\to$ Ka) $\land$ Ka) $\to$ Ha) it's invalid.
b) The formula Hb $\to$ ($\lnot$ Hb $\to$ Kb) it's valid.
a) Ha = true and Ka = true so it is true.
Ha = true and Ka = false so it is true.
Ha = false and Ka = true so is false.
Ha = true and Ka = true so it is true.
So, it's invalid.
b) Hb = true and Kb = true so it is true.
Hb = true and Kb = false so it is true.
Hb = false and Kb = true so it is true.
Hb = true and Kb = true so it is true.
So, it's valid.
I used the truth table to get dos values, sorry I don't know how to build one here, my idea is correct ? The statement a) has an extra parentheses, but I don't know if that is a mistake or on purpose.
